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\begin{document}

\title{7-随机微分方程}
%\institute{上海立信会计金融学院}
\author{{\ppr LQW}}
\renewcommand{\today}{{\ppr \number\year \,年 \number\month \,月 \number\day \,日} }
%\date{{\ppr 2023年1月6日} }

\maketitle

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%\begin{frame}[fragile=singleslide]{1.1.1. }
\begin{frame}{内容提要 }

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\begin{enumerate}
\item[7.2.]  随机微分方程的一般形式
\item[7.4.]  随机微分方程的解的概念
\item[7.5.]  线性的随机微分方程
\item[7.6.]  几何布朗运动对应的随机微分方程
\item[7.7.]  {\ppr Langevin} 方程与 {\ppr Ornstein - Uhlenbeck} 过程
\item[7.8.]  一阶自回归模型
\item[7.10.]  {\ppr Vasicek} 利率模型
\end{enumerate}

\end{frame}

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\begin{frame}{7.1. 确定的微分方程 }

\vspace{-0.4cm}\noindent\makebox[\linewidth]{\rule{\paperwidth}{0.4pt}}
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\begin{itemize}\itemsep0.5em

\item  一阶微分方程是关于自变量 $t$, 应变量 $x(t)$ 以及一阶导数 $x'(t)$ 的等式 $$F(t,x(t),x'(t)) = 0. $$

\item  相比隐式微分方程，我们经常研究显式微分方程 $$x'(t) = f(t,x(t)).$$

\item  当等号右边是可以分离变量的时候，我们可以如下求解
\begin{eqnarray*}
\frac{dx}{dt} = a(t)b(x) \hspace{0.5cm}\Rightarrow \hspace{0.5cm}
\frac{dx}{b(x)} = a(t)dt \hspace{0.5cm}
\Rightarrow \hspace{0.5cm}
\int_{x(0)}^{x(t)} \frac{dx}{b(x)} = \int_0^t a(t)dt.  
\end{eqnarray*}


\end{itemize}

\end{frame}

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\begin{frame}{7.2. 随机的微分方程 }

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\begin{itemize}\itemsep0.5em

\item  微分方程是确定的，未知函数是一个随机过程，初始值是一个随机变量
 $$dX_t = a(t,X_t)dt, \,\,\, X_0(\omega) = Y(\omega).$$

\item  微分方程有一个额外的随机项，未知函数和初始值同上
 $$dX_t = a(t,X_t)dt +b(t,X_t)dB_t, \,\,\, X_0(\omega) = Y(\omega).$$

\item  因为方程里有描述随机扰动的项，所以研究高阶微分方程不是很有意义，为什么？

\item  随机微分方程里的随机过程 $\{B_t\}$ 与 $\{X_t\}$ 都是定义在一个概率空间 $(\Omega, \mathcal{F}, P)$ 与一个递增的事件域序列 $\{\mathcal{F}_t, t\ge 0\}$ 的基础之上的。

\end{itemize}

\end{frame}

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\begin{frame}{7.3. 随机微分方程的严格解释 }

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\begin{itemize}\itemsep0.5em

\item  因为布朗运动 $\{B_t, t\ge 0\}$ 并没有导数，如何理解这个随机微分方程呢？
 $$dX_t = a(t,X_t)dt +b(t,X_t)dB_t. $$

\item  {\color{red}随机微分方程的真实含义是一个关于黎曼积分和伊藤积分的等式，
$$X_t = X_0 + \int_0^t a(s,X_s)ds + \int_0^t b(s,X_s)ds, \,\,\, 0\le t\le T. $$
}

\item  随机过程的伊藤积分是严格定义的，理解为均方收敛。这是包含很多随机变量的某个抽象空间里的一种收敛模式。

\end{itemize}

\end{frame}

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\begin{frame}{7.4. 随机微分方程的强解的概念 }

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\begin{itemize}\itemsep0.5em

\item  一个随机过程 $\{X_t, t\in [0,T]\}$ 称为是随机微分方程 $$dX_t = a(t,X_t)dt +b(t,X_t)dB_t$$ 的一个强解，如果它满足下述三个条件。
\begin{enumerate}
\item  这个随机过程是适应于这个布朗运动的，即 $X_t$ 是 $(B_s,0\le s\le t)$ 的函数。
\item  下述等式中的两个随机积分分别作为黎曼积分和伊藤积分存在，
$$X_t = X_0 + \int_0^t a(s,X_s)ds + \int_0^t b(s,X_s)dB_s, \,\,\, 0\le t\le T. $$
\item  上述等式与 $X_0=Y$ 在概率空间 $(\Omega, \mathcal{F}, P)$ 以概率1成立。
\end{enumerate}

\end{itemize}

\end{frame}

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\begin{frame}{7.5. 线性的随机微分方程 }

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\begin{itemize}\itemsep0.5em

\item  一个线性的随机微分方程由四个常数确定，
$$X_t = X_0 + \int_0^t (c_1X_s+c_2) ds + \int_0^t (\sigma_1X_s + \sigma_2) ds, \,\,\, 0\le t\le T. $$


%\item  

\end{itemize}

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\begin{frame}{7.6. 几何布朗运动 }

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\begin{itemize}\itemsep0.5em

\item  几何布朗运动是指下述随机过程 $$X_t = X_0\exp\left[ (c-\frac{1}{2}\sigma^2)t + \sigma B_t \right], \,\,\, 0\le t\le T.$$ 

\item  几何布朗运动是下述随机微分方程的解 
$$X_t = X_0 + \int_0^t c X_s ds + \int_0^t \sigma X_s ds, \,\,\, 0\le t\le T. $$

\end{itemize}

\end{frame}

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\begin{frame}{7.7. {\ppr Ornstein - Uhlenbeck} 过程 }

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\begin{itemize}\itemsep0.5em

\item  {\ppr Langevin} 方程是指下述随机微分方程 
$$X_t = X_0 + \int_0^t cX_s ds + \int_0^t \sigma dB_s, \,\,\, 0\le t\le T. $$

\item  {\ppr Langevin} 方程的解是下述称为 {\ppr OU} 过程的随机过程
$$X_t = e^{ct}X_0 + \sigma e^{ct}\int_0^t e^{-cs} dB_s, \,\,\, 0\le t\le T. $$


\end{itemize}

\end{frame}

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\begin{frame}{7.8. {\ppr Langevin} 方程与时间序列模型 }

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\begin{itemize}\itemsep0.5em

\item  时间序列是一种离散时间的随机过程。

\item  将 {\ppr Langevin} 方程离散化可以得到一个一阶自回归模型。
\begin{eqnarray*}
dX_t &=& cX_t dt + \sigma dB_t, \\
X_{t+1} - X_t &=& cX_t + \sigma (B_{t+1} - B_t), \\
X_{t+1} &=& \phi X_t + Z_t.
\end{eqnarray*}

\item  其中 $\phi = c+1$ 是常数，$Z_t$ 是独立同分布的 $N(0,\sigma^2)$ 的随机变量序列。

\end{itemize}

\end{frame}

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\begin{frame}{7.9. 计算 {\ppr OU} 过程的均值函数与协方差函数 }

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\begin{itemize}\itemsep0.5em

\item  {\ppr Ornstein - Uhlenbeck} 过程
$$X_t = e^{ct}X_0 + \sigma e^{ct}\int_0^t e^{-cs} dB_s, \,\,\, 0\le t\le T. $$

\item  均值函数、方差函数和协方差函数分别为
\begin{eqnarray*}
\mathbb{E}(X_t) &=& 0, \\
\text{var}(X_t) &=& \frac{\sigma^2}{2c} (e^{2ct}-1), \\
\text{var}(X_s, X_t) &=& \frac{\sigma^2}{2c}\left( e^{c(t+s)} - e^{c(t-s)} \right), \,\,\, s<t. 
\end{eqnarray*}

\end{itemize}

\end{frame}

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\begin{frame}{7.10. {\ppr Vasicek} 利率模型 }

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\begin{itemize}\itemsep0.5em

\item  设 $r_t$ 是即时利率，{\ppr Vasicek} 模型是指下述线性随机微分方程
$$r_t = r_0 + c\int_0^t (\mu-r_s)ds + \sigma \int_0^t dB_s. $$

\item  参数 $c, \mu, \sigma$ 都是正数，各有具体含义。

\item  这个随机微分方程的解为 
$$r_t = r_0e^{-ct} + \mu (1-e^{-ct}) + \sigma e^{-ct} \int_0^t e^{cs}dB_s. $$

\item  在 $\mu=0$ 时，这是 {\ppr Ornstein - Uhlenbeck} 过程。

\end{itemize}

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\begin{thebibliography}{99}

\bibitem{karatzas} {\ppr Ioannis Karatzas, Steven E.  Shreve}. {\ppr Brownian Motion and Stochastic Calculus}. {\ppr GTM 113}. 第{\ppr 2} 版. 世界图书出版公司北京公司，{\ppr 2012} 年 {\ppr 3} 月。

\bibitem{mikosch} {\ppr Thomas Mikosch}. {\ppr Elementary Stochastic Calculus}. 世界图书出版公司，{\ppr 2009} 年 {\ppr 8} 月第 {\ppr 1} 版。
\bibitem{wangjun} 王军、邵吉光、王娟. 随机过程及其在金融领域中的应用. 清华大学出版社，北京交通大学出版社，{\ppr 2018} 年{\ppr 8} 月第 {\ppr 2} 版。
\bibitem{zhangbo} 张波、商豪. 应用随机过程. 中国人民大学出版社，{\ppr 2016} 年 {\ppr 6} 月第 {\ppr 4} 版。
%\bibitem{karlin} {\ppr Mark A. Pinsky, Samuel Karlin}. {\ppr An Introduction to Stochastic Modeling}. 机械工业出版社，{\ppr 2013} 年 {\ppr 2} 月第 {\ppr 1} 版。

\end{thebibliography}

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